Question on limits used in triple integral for volume of a sphere

I am to derive the volume of a sphere using spherical coordinates. I have derived the (correct) jacobian as r^2sin(theta) dr d(theta) d(phi) so its simply a matter of integrating over the correct limits.

2. Relevant equations

What I don't get is why we use 2pi to 0 for phi and pi to 0 for theta, when surely the definition of spherical coordinates uses the opposite limits for the two coordinates.

I am to derive the volume of a sphere using spherical coordinates. I have derived the (correct) jacobian as r^2sin(theta) dr d(theta) d(phi) so its simply a matter of integrating over the correct limits.

2. Relevant equations

What I don't get is why we use 2pi to 0 for phi and pi to 0 for theta, when surely the definition of spherical coordinates uses the opposite limits for the two coordinates.

It depends on your definition of the coordinates. From your above volume element, I presume you're using the convention that the theta angle is the "polar angle" i.e. the one measuring the angle from the z axis, and the phi angle is the "azimuthal angle" i.e. the angle measured in the x-y plane, measured from the x axis.

If these are your conventions, then the theta angle measures from +z to -z (i.e. between 0 and pi) whilst the phi angle covers the whole circle in the x-y plane (i.e. 0 to 2 pi)

Note that there are different conventions used; the convention I mention is used by most applied mathmaticians, engineers, physicists. However, pure mathematicians use the convention that the theta angle is the azimuthal angle, and the phi angle is the polar angle.